Large-amplitude coupled scale-dependent behaviour of geometrically imperfect NSGT nanotubes

In this paper, a scale-dependent coupled nonlinear continuum-based model is developed for the mechanical behaviour of imperfect nanoscale tubes incorporating both the effect of the stress nonlocality and strain gradient effects. The scale effects on the nonlinear mechanics are taken into consideration employing a modified elasticity theory on the basis of a refined combination of Eringen's elasticity and the strain gradient theory. According to the Euler–Bernoulli theory of beams, the nonlocal strain gradient theory (NSGT) and Hamilton's principle, the potential energy, kinetic energy and the work performed by harmonic loads are formulated, and then the coupled scale-dependent equations of the imperfect nanotube are derived. Finally, Galerkin's scheme, as a discretisation technique, and the continuation method, as a solution procedure for ordinary differential equations, are used. The effects of geometrical imperfections in conjunction with other nanosystem parameters such as the nonlocal coefficient as well as the strain gradient coefficient on the coupled large-amplitude mechanical behaviour are explored and discussed.